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    Re: Precision of lunars
    From: George Huxtable
    Date: 2007 Apr 17, 16:49 +0100

    First, I must congratulate Alex on having (astronomically speaking) the most
    precisely-navigated balcony in the whole state of Illinois. Indeed, he has
    shown that the tectonic plate, on which it resides, remains remarkably
    stable, according to the Sun, Moon, and stars..
    
    Seriously though, he has produced a remarkable set of observations, that
    show just what can be done by a careful navigator, in the very best
    conditions of on-land measurement.
    
    He won't mind, I hope, if I add a few comments of my own about his recent
    postings on precise observation.
    
    In Navlist 2567, he wrote, about the combined effect of one-tenth of a
    minute from various sources-
    
    "So you see all these plus-minus 0.1 errors usually cancel each other nicely
    in the process. And you can really use the almanac data for the Lunars and
    obtain very precise results."
    
    What I suggest is that "usually cancel each other nicely" gives quite a
    misleading picture. That seems to have been deduced from a single example,
    in which they happened to cancel; not "usually". True, all the possible
    errors seldom conspire to add together, all in the same direction, to end up
    with maximum-possible error; though that will happen occasionally. Each
    independent source of error, that you consider, does its bit to increase the
    resulting spread, though they add together quadratically, as
    root-sum-of-squares, not linearly.
    
    There are so many quantities (14, according to my posting in Navlist 1417)
    that need to be taken from a modern  Almanac in working a lunar, each of
    which is given  with an error of up to +/- .05', if no other errors exist
    there. Those errors can sum up, then, to a possible, but most unlikely,
    total of 0.7' either way. But statistically speaking, only about 1 in 3 of
    the results will stray beyond +/- 0.1' from that cause, and only 1 in 10
    will be outside 0.2'.
    
    So a significant amount of mutual cancelling is indeed going on, between
    these various almanac errors, which seems to be the point that Alex was
    making. My quibble is only with the way he phrased it.
    
    ========================
    
    In Navlist 2574, Alex quoted a favourite guru of mine, Lord Kelvin, in a
    perceptive account of the precision available from a lunar. That lecture,
    given in 1875 before he was Lord Kelvin, as Sir William Thomson, shows the
    sticking power of audiences and lecturers of those days. How many of us
    would be prepared to sit through, or give, a lecture which occupied, in
    print, 137 (smallish) pages?
    
    Kelvin knew what he was talking about, as a physicist with a serious
    interest in tackling practical problems of his day. He had understood the
    reasons for magnetic compass deviations in steel ships, and developed
    correction procedures which are used in binnacles to this day. He was a
    valued consultant to the many telegraph-laying enterprises of his day, in
    which precise navigation was vital. And he became owner, and certificated
    master, of his own large sailing yacht, the "Lalla Rookh", in which he did
    much ocean voyaging.
    
    Alex points out that his own lunar observations correspond well with
    Kelvin's opinions. In Kelvin's day the astronomical data had to come from
    the nautical almanac, where then, lunar distances were conveniently
    tabulated, at 3-hour intervals, to the nearest arc-second. I don't know how
    good Moon predictions were in the 1870s, but my guess would be to within a
    couple of arc-seconds. Many of the errors involved in predicting a lunar
    distance from the approximations of today's  almanac (which is really
    inconvenient for that purpose) were thus avoided, just as they are by Alex
    when he uses a precise lunar distance from modern computer predictions. Then
    in Kelvin's day, 6-figure log-trig tables were available, so if there were
    no blunders, arithmetical errors would be negligible; just as they are for
    Alex today. It was all down to the skill of the observer, and the precision
    of his sextant, which was comparable with the instruments of today.
    
    Kelvin was discussing what the best navigators could be trusted to obtain at
    sea, and would be presuming benign sea-conditions when doing so. Alex, from
    a firm footing on his balcony, is working under more benign conditions
    still. Even so, he is comparing his own performance with the most expert of
    the men who did that job every day for a living, on which their lives, and
    those of their crew, depended. Looking at his observations, I think he has
    done remarkably well.
    
    Alex discusses his observation, putting them into groups , which have
    different amounts of error from his known position, ignoring the sign of
    that error. I suggest that it may be more informative to group the 85 shots,
    and then plot them, in a way which takes account of that sign, like this-
    
     0.8  1
     0.7  0
     0.6  2
     0.5  2
     0.4  5
     0.3  8
     0.2 12
     0.1 18
     0.0 13
    -0.1  9
    -0.2  7
    -0.3  3
    -0.4  0
    -0.5  5
    
    In which "0.0" represents the range of difference between -0.05 and +0.05,
    and so on. This shows its symmetry well, and it can be compared with a
    standard Gaussian curve.
    
    It can also be expressed as an integrated Gaussian, by plotting an s-curve,
    as follows-
    
     0.8  85
     0.7  85
     0.6  84
     0.5  82
     0.4  80
     0.3  75
     0.2 67
     0.1 55
     0.0 37
    -0.1 24
    -0.2 15
    -0.3  8
    -0.4  5
    -0.5  5
    
    However, these are only different ways of portraying the scatter, and don't
    affect Alex's conclusions.
    
    It's a set of data to be proud of.
    
    Alex discusses in another mailing the positions that Cook achieved. That
    deserves a separate reply, and will get it.
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    
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